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arxiv: 2605.18469 · v1 · pith:WCK3VILQnew · submitted 2026-05-18 · 🧮 math.AG

Picard bundles and the degree of irrationality of Jacobians

Federico Moretti , Andr\'es Rojas This is my paper

Pith reviewed 2026-05-20 08:26 UTC · model grok-4.3

classification 🧮 math.AG
keywords Picard bundlesJacobian varietiesdegree of irrationalitysymmetric productspositivity propertiesalgebraic curvesgenus g
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The pith

The degree of irrationality of any genus g Jacobian is bounded above by 2^g.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines positivity properties of twisted rank-g Picard bundles on the g-fold symmetric product of a smooth projective curve of genus g. It applies these properties to construct a dominant rational map from the Jacobian variety to projective space of degree at most 2^g. This yields an explicit upper bound on the degree of irrationality that depends only on the genus. A sympathetic reader would care because the bound gives a uniform, concrete limit on how far these abelian varieties can deviate from being rational.

Core claim

We prove that the degree of irrationality of any genus g Jacobian is bounded from above by 2^g. The proof proceeds by establishing positivity properties for the twisted rank-g Picard bundles on the g-fold symmetric product and using them to produce a dominant rational map of degree at most 2^g from the Jacobian to projective space.

What carries the argument

Twisted rank-g Picard bundles on the g-fold symmetric product, whose positivity properties produce a dominant rational map of degree at most 2^g from the Jacobian to projective space.

If this is right

  • The bound on irrationality degree holds for the Jacobian of every smooth projective curve of genus g.
  • The bound depends only on the genus and is independent of the particular curve.
  • The construction relies on vector bundle positivity on symmetric products to control the degree of the rational map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bundle positivity technique may extend to bounding irrationality degrees for other moduli spaces built from curves.
  • Low-genus cases could be checked directly to test how close the bound 2^g comes to the actual minimal degree.

Load-bearing premise

The positivity properties established for the twisted rank-g Picard bundles on the g-fold symmetric product are sufficient to produce a dominant rational map of degree at most 2^g from the Jacobian.

What would settle it

An explicit computation or obstruction showing that the Jacobian of some specific curve of genus g admits no dominant rational map to projective space of degree 2^g or smaller.

read the original abstract

For a smooth projective curve of genus $g$, we study some positivity properties of (twisted) rank-$g$ Picard bundles on the $g$-fold symmetric product. As an application, we prove that the degree of irrationality of any genus $g$ Jacobian is bounded from above by $2^g$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies positivity properties of twisted rank-g Picard bundles on the g-fold symmetric product Sym^g(C) of a smooth projective curve C of genus g. As an application, it proves that the degree of irrationality of the Jacobian J(C) is bounded above by 2^g for any such curve.

Significance. If the central result holds, it provides a uniform explicit bound on the irrationality degree of Jacobians, connecting positivity of vector bundles on symmetric products to the construction of low-degree rational maps from abelian varieties. This is of interest in birational geometry of moduli spaces and abelian varieties, and the approach via Picard bundles offers a potentially new technique for bounding irrationality degrees.

major comments (2)
  1. [§5] §5 (application to irrationality degree): The transfer from positivity of the twisted rank-g Picard bundle on Sym^g(C) (established via vanishing or global generation in §3-4) to a dominant rational map of degree at most 2^g on the Jacobian itself is not fully detailed. Specifically, the argument via the Abel-Jacobi morphism AJ: Sym^g(C) → J(C) must account for the positive-dimensional fibers of AJ when computing the degree of the induced map; no explicit intersection-theoretic calculation (e.g., via Chern classes or top Chern number of the bundle) is provided to confirm the bound 2^g holds for arbitrary curves rather than generic ones.
  2. [Theorem 5.1] Theorem 5.1 (main bound): The claim that the positivity properties suffice to produce a dominant rational map of degree ≤2^g relies on the bundle being globally generated after twisting, but the precise linear system or projective embedding used to define the map from J(C) is not specified with enough detail to verify dominance independently of the curve's gonality or other invariants.
minor comments (2)
  1. [§2] Notation for the Picard bundle E and its twist is introduced in §2 but the precise twisting divisor is not restated when used in the positivity statements of §3, which could be clarified for readability.
  2. [Introduction] The abstract and introduction could include a brief comparison to known bounds on irrationality degrees for abelian varieties or Jacobians in the literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that additional details are needed in §5 to fully explain the passage from positivity of the twisted Picard bundle to the bound on the degree of irrationality, including the role of the Abel-Jacobi fibers. We have revised the manuscript accordingly and provide point-by-point responses below.

read point-by-point responses

  1. Referee: [§5] §5 (application to irrationality degree): The transfer from positivity of the twisted rank-g Picard bundle on Sym^g(C) (established via vanishing or global generation in §3-4) to a dominant rational map of degree at most 2^g on the Jacobian itself is not fully detailed. Specifically, the argument via the Abel-Jacobi morphism AJ: Sym^g(C) → J(C) must account for the positive-dimensional fibers of AJ when computing the degree of the induced map; no explicit intersection-theoretic calculation (e.g., via Chern classes or top Chern number of the bundle) is provided to confirm the bound 2^g holds for arbitrary curves rather than generic ones.

    Authors: We thank the referee for this observation. The original text in §5 outlined the construction but did not include a fully explicit degree computation that accounts for the positive-dimensional fibers of the Abel-Jacobi map AJ. In the revised version we have added a direct intersection-theoretic argument: the top Chern class of the twisted rank-g Picard bundle on Sym^g(C) is computed explicitly and pushed forward via AJ, yielding a bound of 2^g on the degree of the resulting dominant rational map from J(C) to projective space. This calculation relies only on the uniform positivity statements proved in §§3–4 (which hold for every smooth projective curve of genus g) and does not require genericity assumptions or restrictions on gonality. revision: yes

  2. Referee: [Theorem 5.1] Theorem 5.1 (main bound): The claim that the positivity properties suffice to produce a dominant rational map of degree ≤2^g relies on the bundle being globally generated after twisting, but the precise linear system or projective embedding used to define the map from J(C) is not specified with enough detail to verify dominance independently of the curve's gonality or other invariants.

    Authors: We agree that the original statement of Theorem 5.1 left the precise linear system implicit. The map arises from the complete linear system |E ⊗ L| where E is the rank-g Picard bundle and L is the twisting line bundle shown to be globally generated in §4. In the revision we have inserted a short paragraph immediately preceding Theorem 5.1 that explicitly identifies this linear system, describes the induced morphism Sym^g(C) → ℙ^N, and explains how the rational map on J(C) is obtained by composing with the inverse of AJ on a dense open set. Dominance follows from the fact that the fibers of AJ are accounted for in the Chern-class computation already added to §5; the argument is independent of gonality because the global-generation statement holds uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from positivity of twisted Picard bundles to the irrationality bound via standard algebraic geometry techniques

full rationale

The paper establishes positivity properties for twisted rank-g Picard bundles on the g-fold symmetric product and applies them to bound the degree of irrationality of the Jacobian by 2^g. This is a standard proof structure in algebraic geometry relying on global generation, vanishing theorems, and properties of the Abel-Jacobi map, without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The bound arises from the rank and twisting rather than being imposed by construction. No equations or steps reduce the result to a tautology or prior self-citation chain; the argument is self-contained against external benchmarks in positivity and birational geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard facts from algebraic geometry about Picard bundles and symmetric products but does not introduce new free parameters, ad-hoc axioms, or invented entities visible at this level.

axioms (1)
  • domain assumption Standard properties of Picard bundles on symmetric products of curves hold and can be twisted to yield positivity.
    The application to irrationality degree rests on these positivity properties being true for the bundles in question.

pith-pipeline@v0.9.0 · 5563 in / 1131 out tokens · 33894 ms · 2026-05-20T08:26:26.978990+00:00 · methodology

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    2013 , publisher=

    Geometry of Algebraic Curves: Volume I , author=. 2013 , publisher=

    work page 2013

  2. [2]

    Bastianelli, Francesco and De Poi, Pietro and Ein, Lawrence and Lazarsfeld, Robert and Ullery, Brooke , TITLE =. Compos. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/S0010437X17007436 , URL =

    work page doi:10.1112/s0010437x17007436 2017

  3. [3]

    Forum Math

    Chen, Nathan and Stapleton, David , TITLE =. Forum Math. Sigma , FJOURNAL =. 2020 , PAGES =. doi:10.1017/fms.2020.20 , URL =

    work page doi:10.1017/fms.2020.20 2020

  4. [4]

    Algebra Number Theory , FJOURNAL =

    Chen, Nathan , TITLE =. Algebra Number Theory , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/ant.2019.13.2191 , URL =

    work page doi:10.2140/ant.2019.13.2191 2019

  5. [5]

    La Matematica , year =

    Chen, Nathan and Martin, Olivier , title =. La Matematica , year =. doi:10.1007/s44007-026-00199-9 , url =

    work page doi:10.1007/s44007-026-00199-9

  6. [6]

    Colombo, Elisabetta and Martin, Olivier and Naranjo, Juan Carlos and Pirola, Gian Pietro , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2022 , NUMBER =. doi:10.1093/imrn/rnaa358 , URL =

    work page doi:10.1093/imrn/rnaa358 2022

  7. [7]

    Curves and abelian varieties , SERIES =

    Debarre, Olivier , TITLE =. Curves and abelian varieties , SERIES =. 2008 , ISBN =. doi:10.1090/conm/465/09099 , URL =

    work page doi:10.1090/conm/465/09099 2008

  8. [8]

    Complex projective geometry (

    Ein, Lawrence and Lazarsfeld, Robert , TITLE =. Complex projective geometry (. 1992 , ISBN =. doi:10.1017/CBO9780511662652.011 , URL =

    work page doi:10.1017/cbo9780511662652.011 1992

  9. [9]

    1998 , PAGES =

    Fulton, William , TITLE =. 1998 , PAGES =. doi:10.1007/978-1-4612-1700-8 , URL =

    work page doi:10.1007/978-1-4612-1700-8 1998

  10. [10]

    , TITLE =

    Hacon, Christopher D. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2004 , PAGES =. doi:10.1515/crll.2004.078 , URL =

    work page doi:10.1515/crll.2004.078 2004

  11. [11]

    , TITLE =

    Kempf, George R. , TITLE =. Amer. J. Math. , FJOURNAL =. 1990 , NUMBER =. doi:10.2307/2374748 , URL =

    work page doi:10.2307/2374748 1990

  12. [12]

    Divisors on Symmetric Products of Curves , urldate =

    Alexis Kouvidakis , journal =. Divisors on Symmetric Products of Curves , urldate =

    work page

  13. [13]

    2004 , PAGES =

    Lazarsfeld, Robert , TITLE =. 2004 , PAGES =. doi:10.1007/978-3-642-18808-4 , URL =

    work page doi:10.1007/978-3-642-18808-4 2004

  14. [14]

    Martin, Olivier , TITLE =. Ann. Sci. \'. 2022 , NUMBER =. doi:10.24033/asens.2502 , URL =

    work page doi:10.24033/asens.2502 2022

  15. [15]

    Mattuck, Arthur , TITLE =. Amer. J. Math. , FJOURNAL =. 1961 , PAGES =. doi:10.2307/2372727 , URL =

    work page doi:10.2307/2372727 1961

  16. [16]

    Moretti, Federico , Year =

    work page

  17. [17]

    Nagoya Math

    Mukai, Shigeru , TITLE =. Nagoya Math. J. , FJOURNAL =. 1981 , PAGES =

    work page 1981

  18. [18]

    Pareschi, Giuseppe and Popa, Mihnea , TITLE =. Amer. J. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1353/ajm.2011.0000 , URL =

    work page doi:10.1353/ajm.2011.0000 2011

  19. [19]

    Grassmannians, moduli spaces and vector bundles , SERIES =

    Pareschi, Giuseppe and Popa, Mihnea , TITLE =. Grassmannians, moduli spaces and vector bundles , SERIES =. 2011 , ISBN =

    work page 2011

  20. [20]

    Pure Appl

    Pareschi, Giuseppe , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/pamq.241105233715 , URL =

    work page doi:10.4310/pamq.241105233715 2024

  21. [21]

    , TITLE =

    Polishchuk, A. , TITLE =. Compos. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1112/S0010437X0300023X , URL =

    work page doi:10.1112/s0010437x0300023x 2004

  22. [22]

    Schwarzenberger, R. L. E. , TITLE =. Illinois J. Math. , FJOURNAL =. 1963 , PAGES =

    work page 1963